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Functions




A relation is any set of ordered pairs.

A function is a set of ordered pairs where no two first values are the same.

To determine whether a set of ordered pairs is a function or a relation:

  1. Look at the first numbers in the ordered pairs

  2. If all of the first numbers are different, then the set of ordered pairs is a function

  3. If any two first numbers are the same, then the set of ordered pairs is a relation

Examples: Identify each set of ordered pairs as a relation or a function. List the domain and range.  

Reasoning


  1. {(-4,3),(5,6),(-4,8),(5,-3)}

    Relation

    Domain: {-4,5}

    Range: {-3,3,6,8}

    Look at the first numbers or the

    x-values in the ordered pairs

    Since there are two -4s, and two 5s, the set of ordered pairs is a relation

    List the domain or x-values

    as {-4,5} because we list each number just once and in ascending order

    List the range or y-values

    as {-3,3,6,8} because we list the numbers in ascending order
    2.  
     
  2. {(-2,6),(-1,5),(0,6),(1,5)}

    Function

    Domain: {-2,-1,0,1}

    Range: {5,6}

    Look at the first numbers or the

    x-values in the ordered pairs

    Since all the x-values are different, the set of ordered pairs is a function

    List the domain or x-values

    as {-2,-1,0,1} because we list the numbers in ascending order

    List the range or y-values

    as {5,6} because we list each number just once and in ascending order


 




Graphic representation of a function and a relation

We can also determine whether a graph represents a relation or a function by looking at the graph’s ordered pairs. In fact, we can tell whether a graph is a function or a relation by just looking at the graph. This is called the vertical line test.

If a vertical line passes through more than one point of a graph, then it is a relation.

If a vertical line does not pass through more than one point of a graph, then it is a function.

Reasoning:  All ordered pairs on the same vertical line have the same x-value.


Example: Identify each graph as a function or a relation.

1.
 

Relation
 

Reasoning
  The graph is a relation because we can draw a vertical line through more than one point
   
2.
 

Function
 

Reasoning
  The graph is a function because any vertical line would pass through only one point

 

Function values of a function

Functions can also be expressed as equations. Because of that, we can find values of functions.
The equations that we will work with will be of the form

y = ax + b and y = ax2 + bx + c.

When we are asked to find the value of a function, we express the functions as f(x) = ax + b and f(x) = ax2 + bx + c, where f(x) = y and tells us to find the value of the function (y) for the given value of x.

f(x) is read "f of x".

Examples: If f(x) = 2x - 5, find:
 
 

Reasoning


  1. f(2)

    f(2) = 2(2) - 5

    f(2) = 4 - 5

    f(2) = -1

    f(2) tells us to substitute 2 in for x and then simplify

    In 2(2) - 5, we multiply first to

    get 4 - 5 then add to get -1

    2.  
  2. f(-3)

    f(-3) = 2(-3) - 5

    f(-3) = -6 - 5

    f(-3) = -11

    f(-3) tells us to substitute -3 in for x and then simplify

    In 2(-3) - 5, we multiply first to

    get -6 - 5 and then add to get -11
 




By finding the values of the function, we can express a set of ordered pairs because f(x) = y.

Since f(2) = -1, y = -1 for (2,-1)

Since f(-3) = -11, y = -11 for (-3,-11)

Remember:  When raising a number to a power, the exponent tells us the sign of our answer. If the exponent is even, then the answer is positive. If the exponent is odd, then the answer is negative.

Example: If f(x) = x2 - 3x + 2, find
 

Reasoning


  1. f(-1)

    f(-1) = (-1)2 - 3(-1) + 2

    f(-1) = 1 + 3 + 2

    f(-1) = 6

    f(-1) tells us to substitute -1 in for x

    In (-1)2 - 3(-1) + 2, order of operations tells us to do the power and multiplication first 1 + 3 + 2 and then add to get 6

     
  2. f(4)

    f(4) = (4)2 - 3(4) + 2

    f(4) = 16 - 12 + 2

    f(4) = 6

    f(4) tells us to substitute 4 in for x and then simplify

    In (4)2 - 3(4) + 2, order of operations tells us to do the power and multiplication first 16 - 12 + 2 and then add to get 6
 

 

Determining the ordered pair solution of a function

A solution of a function is an ordered pair that makes a true sentence. As a result, we can tell whether an ordered pair is a solution by substituting the ordered pair into the equation and determining whether it makes a true sentence.

Example: Determine whether the following ordered pairs is a solution to y = 3x - 5:
 
 

Reasoning


  1. (-3,4)

    4 = 3(-3) - 5

    4 = -9 - 5

    4 ≠ -14

    (-3,4) is not a solution

    Substitute the 4 in for y and

    the -3 in for x

    Simplify by multiplying first

    3(-3) - 5 = -9 - 5 then adding

    -9 - 5 = -14

    4 is not equal to -14, so (-3,4) is not a solution


  2. 4,7)

    7=3(4)-5

    7=12-5

    7=7

    (4,7) is a solution

    substitute the 7 in for y and

    the 4 in for x

    Simplify by multiplying first

    3(4) - 5 = 12 - 5 then adding

    12 - 5 = 7

    7 is equal to 7, so (4,7) is a solution
     

 

 

Try these problems:

Identify if each set of ordered pairs is a relation or a function. List the domain and range.

1. {(-3,8),(-2,5),(3,7),(5,-1)}
2. {(4,3),(5,-2),(4,-4),(5,7)}
 

Identify the following graphs as a relation or a function.
3.
4.
5.
6.
   

 

 

Identify if each set of ordered pairs is a relation or a function. List the domain and range.
1.

{(-3,8),(-2,5),(3,7),(5,-1)}

2.

{(4,3),(5,-2),(4,-4),(5,7)}

 

Identify the following graphs as a relation or a function.
3.

4.

5.

Do the following tables represent a relation or a function? Express the table as a set of ordered pairs. List the domain and range.
6.
7.

Problem Solving
8.
 If you have pieces of paper in one box with the numbers 0, 1, 2, 3, 4 and pieces of paper in a second box with the numbers 5, 6, 7, 8, 9, explain how you could form five ordered pairs that would represent a function. Explain your reasoning.


 

Answers to Practice Problems

  1. Function
    Domain: {-3,-2,3,5}
    Range: {-1,5,7,8}
    2.  
  2. Relation
    Domain: {4,5}
    Range: {-4,-2,3,7}
    3.  
  3. Function
    4.  
  4. Relation
    5.  
  5. Relation
    6.  
  6. Function
    7.  
  7. Relation
    {(-5,7),(3,7),(-5,2),(3,2)}
    Domain: {-5,3}
    Range: {2,7}
    8.  

  8. To make sure that I would have five ordered pairs that represent a function,
    I would draw a number from the first box and then a number from the second
    box and pair them together. I would continue to do this until all five numbers
    from the first box are paired with a number from the second box.  
     

 

 

Find the value of the function f(x) = 2x - 5 for:
1.

f(-2) =

2.

f(0) =

 

Find the value of the function f(x) = 2x2 - 4x + 7 for:
4. f(-2) =
5. f(0) =

 

Determine if each ordered pair is a solution for y = -2x + 3:
7. A = (-2,7)
8. B = (3,6)

 

Determine if each ordered pair is a solution for y = x2 - 2x + 3:
  D = (1,2)
  E = (-2,11)

Answers to Practice Problems
 

  1. f(-2) = 2(-2) - 5
    f(-2) = -4 - 5
    f(-2) = -9
    2.  
  2. f(0) = 2(0) - 5
    f(0) = 0 - 5
    f(0) = -5
    3.  
  3. f(-2) = 2(-2)2 - 4(-2) + 7
    f(-2) = 2(4) + 8 + 7
    f(-2) = 8 + 8 + 7
    f(-2) = 23
    4.  
  4. f(0) = 2(0)2 - 4(0) + 7
    f(0) = 2(0) + 0 + 7
    f(0) = 0 + 0 + 7
    f(0) = 7



    5. Determine if each ordered pair is a solution for y = -2x + 3:


  5. A = (-2,7)
    7 = -2(-2) + 3
    7 = 4 + 3
    7 = 7
    (-2,7) is a solution
    6.  
  6. B = (3,6)
    6 = -2(3) + 3
    6 = -6 + 3
    6 ≠ -3
    (3,6) is not a solution
    7.  

    Determine if each ordered pair is a solution for y = x2 - 2x + 3:


  7. D = (1,2)
    2 = (1)2 - 2(1) + 3
    2 = 1 - 2 + 3
    2 = 2
    (1,2) is a solution
    8.  
  8. E = (-2,11)
    11 = (-2)2 - 2(-2) + 3
    11 = 4 + 4 + 3
    11 = 11
    (-2,11) is a solution
 





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